Order Convergent

In mathematics, specifically in order theory and functional analysis, a filter $\mathcal$ in an order complete vector lattice $X$ is order convergent if it contains an order bounded subset (that is, is contained in an interval of the form $$, b:= \) and if $\mathcal,$
$\sup \left\ = \inf \left\,$
where $\operatorname(X)$ is the set of all order bounded subsets of ''X'', in which case this common value is called the order limit of $\mathcal$ in $X.$

Order convergence plays an important role in the theory of vector lattices because the definition of order convergence does not depend on any topology.

Definition

A net $\left(x_\right)_$ in a vector lattice $X$ is said to decrease to $x_0 \in X$ if $\alpha \leq \beta$ implies $x_ \leq x_$ and $x_0 = inf \left\$ in $X.$
A net $\left(x_\right)_$ in a vector lattice $X$ is said to order-converge to $x_0 \in X$ if there is a net $\left(y_\right)_$ in $X$ that decreases to $0$ and satisfies $\left, x_ - x_0\ \leq y_$ for all $\alpha \in A$.

Order continuity

A linear map $T : X \to Y$ between vector lattices is said to be order continuous if whenever $\left(x_\right)_$ is a net in $X$ that order-converges to $x_0$ in $X,$ then the net $\left(T\left(x_\right)\right)_$ order-converges to $T\left(x_0\right)$ in $Y.$
$T$ is said to be sequentially order continuous if whenever $\left(x_n\right)_$ is a sequence in $X$ that order-converges to $x_0$ in $X,$then the sequence $\left(T\left(x_n\right)\right)_$ order-converges to $T\left(x_0\right)$ in $Y.$

Related results

In an order complete vector lattice $X$ whose order is regular, $X$ is of minimal type if and only if every order convergent filter in $X$ converges when $X$ is endowed with the order topology.

See also

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References

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{{Functional analysis

Functional analysis
Order theory